Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3508 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,16,20,17,21,24,19)(15,18,22,26,23,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,21,27)(16,22,24,28)(19,25,20,26);(1,2,3,4)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27);(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ghi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,g^c=gi,h^c=hi122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300941);